Trigonometric
Integrals
Lesson 8.3
Recall Basic Identities
• Pythagorean Identities
sin 2   cos 2   1
tan 2   1  sec 2 
1  cot   csc 
2
2
• Half-Angle Formulas
1  cos 2
sin  
2
1  cos 2
2
cos  
2
2
These will be used
to integrate powers
of sin and cos
Integral of sinn x, n Odd
• Split into product of an even and sin x
5
4
sin
x
dx

sin

 x  sin x dx
• Make the even power a power of sin2 x
 sin
x  sin x dx    sin x  sin x dx
4
2
2
• Use the Pythagorean identity
  sin x 
2
2
sin x dx   1  cos x  sin x dx
2
2
• Let u = cos x, du = -sin x dx
 1  u

2 2
du    1  2u  u du  ...
2
4
Integral of sinn x, n Odd
• Integrate and un-substitute
2 3 1 5
  1  2u  u du  u  u  u  C
3
5
2
1
3
  cos x  cos x  cos5  C
3
5
2
4
• Similar strategy with cosn x, n odd
Integral of sinn x, n Even
• Use half-angle formulas
1  cos 2
sin  
2
2
4
cos
5x dx Change to power of cos2 x
• Try 
2
1

2
cos
5
x
dx

1

cos10
x
 dx


  2 

2
• Expand the binomial, then integrate
Combinations of sin, cos
• General form
Try with
n
x dx
 sinsin x x cos
 cos x dx

m
2
3
• If either n or m is odd, use techniques as
before
 Split the odd power into an even power and




power of one
Use Pythagorean identity
Specify u and du, substitute
Usually reduces to a polynomial
Integrate, un-substitute
Combinations of sin, cos
• Consider
3
2
sin
4
x

cos
4 x dx

• Use Pythagorean
identity
 sin
3
4 x  1  sin 4 x  dx    sin 4 x  sin 4 x  dx
2
• Separate and use sinn
x strategy for n odd
3
5
Combinations of tanm, secn
• When n is even
 Factor out sec2 x
 Rewrite remainder of integrand in terms of
Pythagorean identity sec2 x = 1 + tan2 x
 Then u = tan x, du = sec2x dx
• Try
sec
y

tan
y
dy

4
3
Combinations of tanm, secn
• When m is odd
Note
similar
 Factor out tan x sec x (for
the
du)strategies for
 Use identity sec2 x – 1 =
integrals involving
tan2combinations
x for evenofpowers
cotm x and cscn x
of tan x
 Let u = sec x, du = sec x tan x
• Try the same integral with this strategy
 sec
4
y  tan y dy
3
Integrals of
Even Powers of sec, csc
• Use the identity sec2 x – 1 = tan2 x
• Try  sec 4 3x dx 
 sec 3x  sec 3x dx 
 1  tan 3x  sec 3x dx 
  sec 3x  tan 3x  sec 3x  dx 
2
2
2
2
2
2
1 3
1
tan 3 x  tan 3 x  C
9
3
2
Wallis's Formulas
• If n is odd and (n ≥ 3) then
 /2

0
n 1
2 4 6
cos x dx      
n
3 5 7
n
And … Believe it or not
• If n is even and (n ≥ 2) then
 /2

0
cos n x dx 
These formulas are also valid if
n  1  by sinnx
5 nx is replaced
1 3 cos

    
n 2
2 4 6
Wallis's Formulas
• Try it out …
 /2

0
 /2

0
sin 7 x dx
cos5 x dx
Assignment
• Lesson 8.3
• Page 540
• Exercises 1 – 41 EOO